Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

Abstract

SUMMARYAn s-adaptive finite element procedure is developed for the transient analysis of 2-D solid mechanics problems with material non-linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user-specified tolerances. The spatial error is quantified by the Zienkiewicz-Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third-order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s-adaptive procedure is the use of finite element mesh superposition (s-refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non-linear transient problems since it is faster, simpler and more efficient than traditional h-refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s-adaptive method for quasi-static and transient problems with material non-linearity.

An interface‐enriched generalized FEM for problems with discontinuous gradient fields

“…These enrichment functions are multiplied by the standard Lagrangian shape functions to provide a sparse resulting system of linear equations. It is worth mentioning that, although (5) seems to indicate that all nodes are enriched, this does not have to be the case in general.…”

“…The first term of (5) is similar to the standard FEM approximation except for the fact that Q u i does not, in general, represent the field value at node i because of the presence of the second term in (5), which is associated with the contribution of enrichment functions in evaluating the nodal values of the solution. These enrichment functions are multiplied by the standard Lagrangian shape functions to provide a sparse resulting system of linear equations.…”

“…Several issues are raised by the implementation of the GFEM formulation described by (5). The first issue involves handling the Dirichlet boundary conditions at the enriched nodes of the mesh.…”

3D hierarchical interface-enriched finite element method: Implementation and applications

“…While this method has been adopted as the underlying computational engine in a variety of commercial software programs and widely used for the analysis and computational deign of engineering problems, its efficient implementation for simulating problems with complex and/or evolving morphologies remains a challenge [4][5][6]. The main roots of this challenge are (i) creating a realistic geometrical model of the problem and (ii) generating a conforming finite element (FE) mesh to discretize this virtual model.…”

An interface‐enriched generalized finite element analysis for electromagnetic problems with non‐conformal discretizations

“…The continued evolution of lighter, stronger, and more efficient systems in civil, industrial, and military applications has fostered long-standing drive for developing novel, sophisticated composite materials [1][2][3]. To analyze the structural, thermal, and electromagnetic properties of these heterogeneous materials, the finite element method (FEM) is usually adopted for its strong adaptability to complex geometries and high numerical accuracy [4][5][6][7][8][9][10][11][12][13][14][15]. At the material interfaces where the field normally exhibits C 0 -continuity, the FEM has to resort to meshes that are conformal with the interfaces to yield an accurate representation of the solution.…”